[Mathematically proficient students] are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others.

The jist, if you aren’t the movie-watching type, is that whenever Wisconsin’s football team scores, their mascot has to do push-ups equivalent to Wisconsin’s total score. This screen grab is a useful talking point.

Play the first act, which ends after Maddow announces the final score of the Wisconsin-Indiana game: 83-20. “83 POINTS!”

Ask your students to write down a guess. “How many push-ups do you think Bucky did over the entire game?” Ask them to write down a number they think is too high and too low.

Here’s where it gets interesting. Ask them to write down all the information they’ll need to figure out the answer. That question is controversial even among math teachers at workshops I facilitate. Some argue that all you need to know is that Wisconsin scored eleven touchdowns and two field goals. Others argue that you also need to know the order of those touchdowns and field goals. In W. Stephen Wilson’s ideal math classroom, we’re stuck. Communication is inessential to Wilson’s ideal math classroom but communication is essential to any resolution of this dispute.

The mathematical practice standards require an argument. Both sides aren’t right. How will one side persuade the other? At this point, we learn a useful technique for arguing mathematically. One side has said, “Order never matters.” All we need to sink that rule is a single counterexample. One person suggests trying [7, 7, 3] and then [7, 3, 7] — the same scores in different sequence. Another suggests an even less costly test of [7, 3] and [3, 7]. And the matter is settled.

Having established that order matters, another question then arises: “If you’re Bucky, when do you want your team to score its field goals — at the end of the game or the beginning?”

The question that’s rarely asked in print is, “What information will you need?” That information is generally nailed to the floor, written directly on the page. NCTM has revealed in the text of the problem that the order of the scores matters when all the action is in deciding whether or not the order of the scores matters.

Sidenote #2: Opposition To The CCSS Makes For Very Strange Bedfellows

The CCSS aren’t remotely above criticism. It’s bizarre to me, though, how many edtech pundits leapt on that Fordham piece, grateful for any institutional validation of their position against the CCSS. But Wilson and Wurman, the authors, like the punditry’s technological utopianism even less than they do the CCSS. The enemy of your enemy is not your friend.

My best two students disagreed on whether order mattered and I was able to convince (falsely) one of them that order didn’t matter. And sure enough one of my “average” students – who always works her butt off but is rarely rewarded publicly in class for that work – was the only one to figure out and show that order matters.

19 Comments

Jerrid Kruse

You said, “The question that’s rarely asked in print is, “What information will you need?” That information is generally nailed to the floor, written directly on the page.”

While I agree this is usually the case, & I (as anyone who has ever interacted with me can note) believe different media do have biases. However, there is a delicate line (fuzzy gray area?) between media bias & human bias. I wonder if your claim about print is really about textbook publishers desire to be overly helpful rather than about the medium. That is, I could remove those details from print & I could add them to video.

I think the importance of your ideas goes well beyond the medium we choose & is more about the attitude in which we approach the teaching event. Do we want to ensure success or ensure thought?

The context matters: “There will always be people who believe that you do not understand mathematics if you cannot write a coherent essay about how you solved a problem, thus driving future STEM students away from mathematics at an early age. A fairness doctrine would require English language arts (ELA) students to write essays about the standard algorithms, thus also driving students away from ELA at an early age. The ability to communicate is NOT essential to understanding mathematics.”

The ability to communicate in math is essential to demonstrating an understanding.

You’ve shown that yourself in the item description. You didn’t resort to a long-winded essay detailing the steps you’d take to discovering the method to determining the way to find Bucky’s total.

“All we need to sink that rule is a single counterexample. One person suggests trying [7, 7, 3] and then [7, 3, 7] — the same scores in different sequence.”

A math proof is simply an essay written using a grammar and syntax particular to the topic. Using an alternate form does little to help the situation, much as attempting to explain sub-atomic theory by only using words and theory your five-year-old would understand.

Matt H.

I hope I’m not too off-topic here, but I personally think the more interesting question is, “Given the total number of push-ups Bucky performed during the game, and the rules of NCAA football, what scores are possible?”

Justin

I love the three act math videos, concept and design! I notice that some of the lessons that are located on the Google doc link to video files that must be opened. Others, such as this one, link to a web page with embedded videos. Is there an embedded page for each three act prompt? If so, how can I access the web page instead of just the video file? Thanks!

This makes me think of another question: if the Badgers score x touchdowns in a game, and no field goals, how many pushups does Bucky do? (function in terms of x) But of course excluding field goals is artificial. I wonder if it’s possible to write a mathematical function that accounts for both, and handles order correctly…

What’s the maximum and minimum number of pushups Bucky could have done for that 83 point game?

If you just knew the 83 points, what is the most likely number of pushups? (Do you weight all possibilities equally? Do you look back at all other teams that scored 83 points and use just those combinations? Do you look at all teams that scored exactly 83 at some point during the game and use those combinations?

Does the most likely number end up being the average of all possibilities? The median? The mode? The simple mean of the max and min?

Could you make a function that, given any score, you could give the most likely number of pushups?

@Jerrid: “I wonder if your claim about print is really about textbook publishers desire to be overly helpful rather than about the medium.”

I had a similar thought. Perhaps published curriculum tends to be overly helpful/guided in response to a distrust of the teacher to effectively implement the exploration?

“Do we want to ensure success or ensure thought?”

Beautifully put. This is assuming we define “success” as getting the answer that was intended by the designer of the task. Maybe success should be defined as having accomplished thought (as you put it) or as students simply engaging in and doing mathematics?

I wonder if your claim about print is really about textbook publishers desire to be overly helpful rather than about the medium. That is, I could remove those details from print & I could add them to video.

Most publishers would say (and I’d agree) that the goal state of the problem should be clear and all the tools, information, and resources for reaching it should be within the student’s reach. (They aren’t going to give a student a task that’s impossible to answer.) That means the information will be available somewhere. In print, it’s generally available in the text of the problem itself which makes it very difficult to ask questions like, “What information do you need here?” You could put that information on another page, but the suggestion feels silly. We’d be bending over backwards to accommodate the flaws of the medium. With digital media, I can pose the task over several screens (at no extra expense), each of which could require a response from the student, essentially asking them to work along the entire ladder of abstraction.

Could I mess that up with digital media? Definitely. I’d rather compare the best of what’s possible with these media than the worst, though.

@Mathcurmudgeon, you’re attributing nuances to Wilson that I don’t find in his essay. He’s emphatic about communication — not about a particular genre of communication.

Justin: Is there an embedded page for each three act prompt?

Nope. Some three-act tasks just offer the downloadables. Others have a lesson plan explaining their use. I’ve marked those tasks with a column in the spreadsheet called “Lesson Plan.”

Finally, a nod to everyone who’s suggested alternate questions for the Bucky prompt, some of which were built into the task card already as sequels. But these questions aren’t equally enticing to your students. I often misjudge which angle on a problem appeals most to a student. That’s where I find 101questions useful. Which question do these people find most appealing?

(Cutting this one off at the pass: anybody who recommends we let students take any angle on the problem they want will ideally demonstrate they’ve fully considered the implications of that recommendation.)

mr bombastic

That said, I don’t think that being able to communicate, using mathematical language or verbally, is essential to understanding mathematics. I have noticed a fairly common type of student that has good mathematical thinking, but gets destroyed by the communication piece. They have difficulty with both reading and writing, whether it is English or mathematical notation, but are much better than average thinkers in that they are able to apply prior ideas to new situations. Being able to communicate math and being good at math are both desirable, but they are two different things.

This idea also seems to tie into Devlin’s project of trying to make games that take all of the language out of the mathematical ideas. The mathematical thinking and mathematical communication are two different things.

Of course, all of the above depends on your definition of math and communication.

It will never be possible to dissuade people like Wilson and Wurman to stop parodying what is meant by the communication strand in any of the NCTM standards volumes or in the process standards in the otherwise mostly useless Common Core math pseudo-standards.

They have each been major players on the anti-progressive side of the Math Wars for a couple of decades and have no compunction about lying, distorting the truth, engaging in the most absurd sorts of straw man fallacies and hyperbole to convince as many careless people as possible that any approach to mathematics education that doesn’t look just like what they (as “math” types) received is a harbinger of the Four Horseman of the Educational and Economic Apocalypse.

As far as I know, Ze’ev has never taught, and I doubt Wilson has taught below the elite university level. What they know about teaching math to K-12 kids is essentially nothing.

Spend a day in math classrooms in any high-needs school (or, frankly, not-so-high-needs school) and tell me that it suffices to demonstrate solid mathematical competency and understanding to be able to calculate correctly some routine arithmetic, algebraic, or geometric exercise and “show your work.” Give some of the stronger-performing kids a contextualized problem, say one in which one computes various critical points of a quadratic equation for projectile motion and then ask what they mean IN THE CONTEXT of the problem. You’ll discover that the x-intercept at x = -3 means that the projectile was on the ground precisely at 3 seconds before launch and that it traveled smoothly and continuously along the same trajectory that the projectile follows from release to the second x-intercept. Apparently, no baseball player having a catch has ever juggled a ball for a few seconds before throwing it back.

There are textbooks that deliberately do not include all information needed to solve problems. The physics text I’m using to home-school my son in Physics (Matter and Interactions, by Chabay and Sherwood) fairly routinely expects students to look up facts on the web (like the density of a particular metal, or the mass of a particular planet), without telling them that they need to do so.

Most of the time this is fine, but a couple of times there have been problems for which the theory necessary for any sort of reasonable solution had not been covered (one problem at the end of Chapter 8 was poorly worded, so that it could not be answered without a theory of blackbody radiation, which was not covered anywhere in the book).

Certainly many math problems can be done with incomplete information (resulting in a function of the missing variables, rather than a numeric answer). I don’t think that such word problems are done nearly often enough in math classes.

Re: 3-Act tasks in general. Yesterday while reviewing some trigonometry, specifically trig ratios and the Law of Sines, one student said, “This is the stuff that if we applied it to the real world we could solve the world!” I’ve tried my hand at creating 3-Acts using trig but they come out feeling contrived. So I’m requesting help from Dan or his readers coming up with opportunities for Nat to “solve the world!” or finding ones that already exist.

Tom

Dan, thanks for posting this – I’ve been grabbing your 3acts and modifying them and using them as much as I can – it’s my first year teaching and I’ve never taught Pre-Calc before so it’s been tough getting creative when I’m just trying to stay ahead of the kids .

I used this 3Act almost word for word from the task page and it was great. Students who normally were checked out were completely engaged.

My best two students disagreed on whether order mattered and I was able to convince (falsely) one of them that order didn’t matter. And sure enough one of my “average” students – who always works her butt off but is rarely rewarded publicly in class for that work – was the only one to figure out and show that order matters.

It was the perfect transition away from conic sections and graphing and into discrete math, permutations, and sequences/series.

And the cherry on top – I had asked a few weeks ago to be observed by our senior teacher and today was the day he came in!

Thanks for the feedback, Tom. I’m always interested in how these play out on the ground, particularly the ones that don’t work, or that needed a lot of modification. My comments or e-mail box are always open to you: dan@mrmeyer.com